Not being (super)thin or solid is hard: A study of grid Hamiltonicity

Authors:
Esther M. Arkin;Sándor P. Fekete;Kamrul Islam;Henk Meijer;Joseph S. B. Mitchell;Yurai Núòez-Rodríguez;Valentin Polishchuk;David Rappaport;Henry Xiao
Affiliations:
Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, USA;Institute of Operating Systems and Computer Networks, Department of Computer Science, Braunschweig University of Technology, Germany;School of Computing, Queen's University, Canada;School of Computing, Queen's University, Canada and Department of Science, Roosevelt Academy, The Netherlands;Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, USA;School of Computing, Queen's University, Canada;Helsinki Institute for Information Technology, the University of Helsinki and Helsinki University of Technology, Finland;School of Computing, Queen's University, Canada;School of Computing, Queen's University, Canada
Venue:
Computational Geometry: Theory and Applications
Year:
2009

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Abstract

We give a systematic study of Hamiltonicity of grids - the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, ''square'', grids, we give a comprehensive taxonomy of the grid graphs. For many classes of grid graphs we resolve the computational complexity of the Hamiltonian cycle problem. For graphs for which there exists a polynomial-time algorithm we give efficient algorithms to find a Hamiltonian cycle. We also establish, for any g=6, a one-to-one correspondence between Hamiltonian cycles in planar bipartite maximum-degree-3 graphs and Hamiltonian cycles in the class C"g of girth-g planar maximum-degree-3 graphs. As applications of the correspondence, we show that for graphs in C"g the Hamiltonian cycle problem is NP-complete and that for any N=5 there exist graphs in C"g that have exactly N Hamiltonian cycles. We also prove that for the graphs in C"g, a Chinese Postman tour gives a (1+8g)-approximation to TSP, improving thereby the Christofides ratio when g16. We show further that, in any graph, the tour obtained by Christofides' algorithm is not longer than a Chinese Postman tour.